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- W1775636183 abstract "In the framework of <bold>ZF</bold>, i.e., Zermelo-Fraenkel set theory without the axiom of choice <bold>AC</bold>, we show that if the family of all non-empty, closed subsets of a metric space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper X comma d right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(X,d)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a choice function, then so does the family of all non-empty, open subsets of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In addition, we establish that the converse is not provable in ZF. We also show that the statement “<italic>every subspace of the real line <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with the standard topology has a choice function for its family of all closed, non-empty subsets</italic> is equivalent to the weak choice form “<italic>every continuum sized family of non-empty subsets of reals has a choice function</italic>." @default.
- W1775636183 created "2016-06-24" @default.
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- W1775636183 date "2005-06-03" @default.
- W1775636183 modified "2023-09-26" @default.
- W1775636183 title "Weak axioms of choice for metric spaces" @default.
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- W1775636183 doi "https://doi.org/10.1090/s0002-9939-05-07970-0" @default.
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